Proof that rational times irrational is irrational | Algebra I | Khan Academy

TL;DR

Multiplying a rational number by an irrational number results in an irrational number.

Transcript

What I want to do with this video is do a quick proof that if we take a rational number, and we multiply it times an irrational number, that this is going to give us an irrational number. And I encourage you to actually pause the video and try to think if you can prove this on your own. And I'll give you a hint. You can prove it by a proof through ... Read More

Key Insights

• ⌛ The video presents a proof by contradiction to demonstrate that a rational times an irrational is irrational.
• 😑 Expressing the assumed irrational number as the ratio of two integers contradicts the assumption and proves it false.
• 🛀 The proof shows the importance of logical reasoning and using contradiction to establish mathematical truths.
• #️⃣ Understanding the properties of irrational and rational numbers is crucial in comprehending the proof.
• 🏑 This proof has applications in various mathematical fields such as algebra, calculus, and number theory.
• #️⃣ The concept of irrational numbers being multiplied by rational numbers helps establish the real number system's completeness.
• #️⃣ The proof strengthens the understanding of the fundamental properties of rational and irrational numbers.

Questions & Answers

Q: How does the video prove that a rational times an irrational is irrational?

The video uses a proof by contradiction, assuming the product of a rational and an irrational is rational. By expressing the assumed irrational number as the ratio of two integers, it is shown that the assumption leads to a contradiction, proving that a rational times an irrational is irrational.

Q: What is the significance of expressing the assumed irrational number as the ratio of two integers?

Expressing the assumed irrational number as the ratio of two integers shows that it can be represented as a rational number. This contradicts the initial assumption and proves that a rational times an irrational is always irrational.

Q: Why is the irrational number x expressed as mb/na?

The irrational number x is expressed as mb/na to show that both the numerator (mb) and the denominator (na) are integers. This representation further highlights that the assumed irrational number can be expressed as a ratio of two integers, proving it to be rational.

Q: What is the conclusion of the proof?

The conclusion of the proof is that a rational times an irrational is always irrational. The assumption that a rational times an irrational is rational leads to a contradiction, proving the opposite to be true.

Summary & Key Takeaways

• The video aims to prove that multiplying a rational number by an irrational number always produces an irrational number.

• The proof is done through a proof by contradiction, assuming that the product of a rational and an irrational is rational and then showing that it leads to a contradiction.

• By expressing the assumed irrational number as the ratio of two integers, it is shown that the assumption is false and that a rational times an irrational is always irrational.